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8. Proof techniques and examples

    1. Problem 8.1.

      Is is possible to show that \begin{equation*} \int_X^{2X} \left| \sum_{x\leq n\leq x+H} \lambda(n) e(n \alpha) \right| d ̣x = o(XH) \end{equation*} with $H > X^\epsilon$ implies \begin{equation*} \sum_{n \leq X} \frac{\lambda(n)\lambda(n+1)}{n} = o(\log X)? \end{equation*}
        • Problem 8.2.

          Does the Sarnak conjecture hold uniformly in $x \in X$ in the case of horocycle flows? It is known that the uniform Sarnak conjecture follows from the Sarnak conjecture, but we don’t know that it follows case by case.
            • Problem 8.3.

              The Chowla conjecture implies that $\mu$ cannot be orthogonal to all uniquely ergodic systems (allowing non-zero entropy). Can we find a concrete example of a uniquely ergodic system that correlates with $\mu$? A way to find a uniquely ergodic system (of positive entropy) that does not correlate with $\lambda$ is known.
                1. Remark. [Nikos] Since the system of Problem 3.1 is disjoint from all ergodic systems, an example of an ergodic sequence (or uniquely ergodic system) that correlates with the Liouville function would give a negative answer to Problem 3.1.

                      Cite this as: AimPL: Sarnak's conjecture, available at http://aimpl.org/sarnakconjecture.